WASHINGTON—The focus of infrastructure talks shifted Wednesday to a bipartisan group of senators after an initial round of White House discussions broke down, but negotiators face the same hurdles on how to pay for any major new spending package.

Discussions intensified among a group of centrist-leaning Senate Republicans and Democrats trying to strike a deal that could spend up to $900 billion over five years, according to people familiar with the talks. A small group of Republicans involved in those discussions huddled Wednesday to hash out details of their proposal, which they discussed with GOP leaders later in the day. GOP lawmakers didn’t comment on their leaders’ reception to their ideas as they left the meeting.

Sen. Mitt Romney (R., Utah), who is part of the GOP wing of the bipartisan group, said there were eight or nine Republican senators involved in developing the new proposal, which he said wouldn’t include any tax increases. “That’s a red line for Republicans,” he said. So far the bipartisan group has declined to specify how it would pay for an infrastructure proposal.

The fresh effort came a day after President Biden ended an earlier series of talks with a GOP group led by Sen. Shelley Moore Capito (R., W.Va.). Senior Republicans were already signaling skepticism that the bipartisan group would be able to produce a breakthrough that eluded Mr. Biden and Mrs. Capito over weeks of talks.

“I wish them every success if they can find the magic key that unlocks the door,” Sen. John Cornyn of Texas, a member of Senate GOP leadership, said Wednesday of the bipartisan group. But, he added, “there’s nothing that says that whatever this group agrees to that other Republicans are necessarily going to support—to me, that’s the flaw in this approach.”

Senate Majority Leader Chuck Schumer (D., N.Y.) indicated on Tuesday that Democrats were preparing to move forward with at least part of an infrastructure package through a process tied to the budget that would allow legislation to pass with just Democratic support. But some Democrats, most prominently Sen. Joe Manchin (D., W.Va.), have said they don’t want to push through legislation on party lines, at least not until all efforts at reaching a bipartisan deal have been exhausted.

One outcome of the latest round of negotiations could be to persuade Mr. Manchin and Sen. Kyrsten Sinema (D., Ariz.), who are both involved in the talks, that a bipartisan agreement is impossible, paving the way for Democrats to muscle through an infrastructure package without GOP support.

White House press secretary Jen Psaki said Wednesday that Mr. Biden believes that there are multiple paths forward and was encouraged by the bipartisan Senate group, as well as a more fleshed-out proposal from the Problem Solvers Caucus, a bipartisan House group. The Problem Solvers on Tuesday night released a more detailed $1.249 trillion proposal over eight years, of which $762 billion would be new spending above a projected baseline of continued federal spending.

Ms. Psaki also said the White House was pleased to see the Senate moving forward with the budget process and the ability to pass legislation with Democratic support, known as reconciliation. “It’s an opportunity to move through that vehicle a bunch of his bold economic ideas, as well as corporate tax reform,” she said.

The negotiations between Mr. Biden and Mrs. Capito faltered in part because the two sides couldn’t reach any agreement on how to offset the cost of their proposals, which also remained nearly $700 billion apart in size. Republicans rejected Mr. Biden’s call to increase the corporate tax rate, as well as other proposals floated by the White House, which Republicans said amounted to comparable tax increases through different methods.

The White House, meanwhile, said Republicans weren’t willing to spend enough to meet Mr. Biden’s threshold for a sufficiently sweeping infrastructure, clean energy and job creation package.

Mr. Biden also faces pressure from his party’s left wing, which raised concerns Wednesday over the prospect of omitting some of the more ambitious proposals to combat climate change.

“An infrastructure package that goes light on climate and clean energy should not count on every Democratic vote,” Sen. Martin Heinrich (D., N.M.) said on Twitter Wednesday. He was reacting to comments by National Climate Adviser Gina McCarthy to Politico that some climate elements could be left out of the final package.

The White House has, in recent days, pointed to a $547 billion infrastructure bill authored by Democrats on the House Transportation and Infrastructure Committee as one avenue forward.

The package dedicates funding to roads, rail and public transit while also reauthorizing federal transportation programs. But it is much smaller in scope than Mr. Biden’s $2.3 trillion program, offering far less funding for administration priorities like electric vehicles and not addressing swaths of the White House plan, including care for elderly and disabled Americans.

Lawmakers face a Sept. 30 deadline for reauthorizing many federal infrastructure programs, a step Congress must take every few years. Without a broader agreement on infrastructure legislation last year, lawmakers approved a short-term extension of existing programs for one year, and members of both parties say they are eager to craft a longer-term agreement.

Reauthorizing and adjusting transportation programs, which dictate how states can use federal funds on infrastructure projects, may be difficult to accomplish through reconciliation, according to lawmakers and aides. Those expected procedural hurdles have prompted some Democrats, including House Transportation and Infrastructure Committee Chairman Peter DeFazio (D., Ore.), to push for a bipartisan agreement on the reauthorization legislation.

But even an agreement on narrower reauthorization legislation may be difficult to ultimately accomplish, as Republicans lined up against the $547 billion House bill. The Senate Environment and Public Works Committee unanimously approved a $303 billion reauthorization bill, and other Senate committees are working on other elements of infrastructure reauthorization. Lawmakers will have the option of approving another one-year extension of current programs if talks fall apart.

—Ken Thomas contributed to this article.

Write to Kristina Peterson at kristina.peterson@wsj.com and Andrew Duehren at andrew.duehren@wsj.com